Theorem cfinufil 23822

Description: An ultrafilter is free iff it contains the Fréchet filter cfinfil 23787 as a subset. (Contributed by NM, 14-Jul-2008.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Assertion

Ref	Expression
cfinufil	⊢ (𝐹 ∈ (UFil‘𝑋) → (∩ 𝐹 = ∅ ↔ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ Fin} ⊆ 𝐹))

Distinct variable groups:   𝑥,𝐹   𝑥,𝑋

Proof of Theorem cfinufil

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elpwi 4573	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋)
2		ufilb 23800	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (¬ 𝑥 ∈ 𝐹 ↔ (𝑋 ∖ 𝑥) ∈ 𝐹))
3	2	adantr 480	. . . . . . . . 9 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) ∧ (𝑋 ∖ 𝑥) ∈ Fin) → (¬ 𝑥 ∈ 𝐹 ↔ (𝑋 ∖ 𝑥) ∈ 𝐹))
4		ufilfil 23798	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
5	4	adantr 480	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → 𝐹 ∈ (Fil‘𝑋))
6		filfinnfr 23771	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑋 ∖ 𝑥) ∈ 𝐹 ∧ (𝑋 ∖ 𝑥) ∈ Fin) → ∩ 𝐹 ≠ ∅)
7	6	3exp 1119	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (Fil‘𝑋) → ((𝑋 ∖ 𝑥) ∈ 𝐹 → ((𝑋 ∖ 𝑥) ∈ Fin → ∩ 𝐹 ≠ ∅)))
8	7	com23 86	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Fil‘𝑋) → ((𝑋 ∖ 𝑥) ∈ Fin → ((𝑋 ∖ 𝑥) ∈ 𝐹 → ∩ 𝐹 ≠ ∅)))
9	5, 8	syl 17	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → ((𝑋 ∖ 𝑥) ∈ Fin → ((𝑋 ∖ 𝑥) ∈ 𝐹 → ∩ 𝐹 ≠ ∅)))
10	9	imp 406	. . . . . . . . 9 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) ∧ (𝑋 ∖ 𝑥) ∈ Fin) → ((𝑋 ∖ 𝑥) ∈ 𝐹 → ∩ 𝐹 ≠ ∅))
11	3, 10	sylbid 240	. . . . . . . 8 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) ∧ (𝑋 ∖ 𝑥) ∈ Fin) → (¬ 𝑥 ∈ 𝐹 → ∩ 𝐹 ≠ ∅))
12	11	necon4bd 2946	. . . . . . 7 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) ∧ (𝑋 ∖ 𝑥) ∈ Fin) → (∩ 𝐹 = ∅ → 𝑥 ∈ 𝐹))
13	12	ex 412	. . . . . 6 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → ((𝑋 ∖ 𝑥) ∈ Fin → (∩ 𝐹 = ∅ → 𝑥 ∈ 𝐹)))
14	13	com23 86	. . . . 5 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (∩ 𝐹 = ∅ → ((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹)))
15	1, 14	sylan2 593	. . . 4 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → (∩ 𝐹 = ∅ → ((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹)))
16	15	ralrimdva 3134	. . 3 ⊢ (𝐹 ∈ (UFil‘𝑋) → (∩ 𝐹 = ∅ → ∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹)))
17	4	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
18		uffixsn 23819	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → {𝑦} ∈ 𝐹)
19		filelss 23746	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ {𝑦} ∈ 𝐹) → {𝑦} ⊆ 𝑋)
20	17, 18, 19	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → {𝑦} ⊆ 𝑋)
21		dfss4 4235	. . . . . . . . . . 11 ⊢ ({𝑦} ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ {𝑦})) = {𝑦})
22	20, 21	sylib 218	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → (𝑋 ∖ (𝑋 ∖ {𝑦})) = {𝑦})
23		snfi 9017	. . . . . . . . . 10 ⊢ {𝑦} ∈ Fin
24	22, 23	eqeltrdi 2837	. . . . . . . . 9 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → (𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin)
25		difss 4102	. . . . . . . . . . 11 ⊢ (𝑋 ∖ {𝑦}) ⊆ 𝑋
26		filtop 23749	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹)
27		elpw2g 5291	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝐹 → ((𝑋 ∖ {𝑦}) ∈ 𝒫 𝑋 ↔ (𝑋 ∖ {𝑦}) ⊆ 𝑋))
28	17, 26, 27	3syl 18	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → ((𝑋 ∖ {𝑦}) ∈ 𝒫 𝑋 ↔ (𝑋 ∖ {𝑦}) ⊆ 𝑋))
29	25, 28	mpbiri 258	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → (𝑋 ∖ {𝑦}) ∈ 𝒫 𝑋)
30		difeq2 4086	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑋 ∖ {𝑦}) → (𝑋 ∖ 𝑥) = (𝑋 ∖ (𝑋 ∖ {𝑦})))
31	30	eleq1d 2814	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑋 ∖ {𝑦}) → ((𝑋 ∖ 𝑥) ∈ Fin ↔ (𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin))
32		eleq1 2817	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑋 ∖ {𝑦}) → (𝑥 ∈ 𝐹 ↔ (𝑋 ∖ {𝑦}) ∈ 𝐹))
33	31, 32	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑋 ∖ {𝑦}) → (((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹) ↔ ((𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin → (𝑋 ∖ {𝑦}) ∈ 𝐹)))
34	33	rspcv 3587	. . . . . . . . . 10 ⊢ ((𝑋 ∖ {𝑦}) ∈ 𝒫 𝑋 → (∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹) → ((𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin → (𝑋 ∖ {𝑦}) ∈ 𝐹)))
35	29, 34	syl 17	. . . . . . . . 9 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → (∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹) → ((𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin → (𝑋 ∖ {𝑦}) ∈ 𝐹)))
36	24, 35	mpid 44	. . . . . . . 8 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → (∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹) → (𝑋 ∖ {𝑦}) ∈ 𝐹))
37		ufilb 23800	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ {𝑦} ⊆ 𝑋) → (¬ {𝑦} ∈ 𝐹 ↔ (𝑋 ∖ {𝑦}) ∈ 𝐹))
38	20, 37	syldan 591	. . . . . . . . 9 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → (¬ {𝑦} ∈ 𝐹 ↔ (𝑋 ∖ {𝑦}) ∈ 𝐹))
39	18	pm2.24d 151	. . . . . . . . 9 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → (¬ {𝑦} ∈ 𝐹 → ¬ 𝑦 ∈ ∩ 𝐹))
40	38, 39	sylbird 260	. . . . . . . 8 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → ((𝑋 ∖ {𝑦}) ∈ 𝐹 → ¬ 𝑦 ∈ ∩ 𝐹))
41	36, 40	syld 47	. . . . . . 7 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → (∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹) → ¬ 𝑦 ∈ ∩ 𝐹))
42	41	impancom 451	. . . . . 6 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹)) → (𝑦 ∈ ∩ 𝐹 → ¬ 𝑦 ∈ ∩ 𝐹))
43	42	pm2.01d 190	. . . . 5 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹)) → ¬ 𝑦 ∈ ∩ 𝐹)
44	43	eq0rdv 4373	. . . 4 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹)) → ∩ 𝐹 = ∅)
45	44	ex 412	. . 3 ⊢ (𝐹 ∈ (UFil‘𝑋) → (∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹) → ∩ 𝐹 = ∅))
46	16, 45	impbid 212	. 2 ⊢ (𝐹 ∈ (UFil‘𝑋) → (∩ 𝐹 = ∅ ↔ ∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹)))
47		rabss 4038	. 2 ⊢ ({𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ Fin} ⊆ 𝐹 ↔ ∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹))
48	46, 47	bitr4di 289	1 ⊢ (𝐹 ∈ (UFil‘𝑋) → (∩ 𝐹 = ∅ ↔ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ Fin} ⊆ 𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  ∀wral 3045  {crab 3408   ∖ cdif 3914   ⊆ wss 3917  ∅c0 4299  𝒫 cpw 4566  {csn 4592  ∩ cint 4913  ‘cfv 6514  Fincfn 8921  Filcfil 23739  UFilcufil 23793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1o 8437  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-fbas 21268  df-fg 21269  df-fil 23740  df-ufil 23795

This theorem is referenced by: (None)